HI Spammy-7356,
1.What are RDS related roles installed in your environment?
Remote Desktop Services roles
https://learn.microsoft.com/en-us/windows-server/remote/remote-desktop-services/rds-roles
2.From server 2012, remote desktop service architecture has been changed, so we can not migrate from terminal server 2008r2 to terminal server 2016 directly.
3.Since I am not sure if your all old apps which are installed on terminal server 2008r2 is compatible to server 2016/2019, so I suggest that we can build a new terminal servers with OS w2016/w2019 in your environment and install your apps on it then check if these apps can run together with w2016/w2019 fine.
4.Since mainstream End Date of windows server 2016 is 01/11/2022, I suggest that you can build windows server 2019 for your terminal server and install "remote desktop session host" role then install all your apps on it for testing.
Windows Server 2016 follows the Fixed Lifecycle Policy.
https://learn.microsoft.com/en-us/lifecycle/products/windows-server-2016
Server 2019 evaluation download link
https://www.microsoft.com/en-us/evalcenter/evaluate-windows-server-2019
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